jean-pierre serre - johann bernoulli institute for ...top/lectures/delft.pdf · jean-pierre serre:...
TRANSCRIPT
born 15 September 1926
PhD in 1951 (“Homologie singuliere des espaces fibres. Appli-cations”)
supervisor: Henri Cartan (Sorbonne, Paris)
1956–1994 professor in Algebra & Geometry at the College deFrance (Paris)
Collected papers (4 volumes) contain 173 items (including manyletters and abstracts of courses given at the College de France)
13 books
Most recent text in Collected papers: 1998.
Most recent text according to MathSciNet: 2006.
3
many distinctions
honorary degrees from Cambridge, Stockholm, Glasgow, Athens,
Harvard, Durham, London, Oslo, Oxford, Bucharest, Barcelona
honorary member or foreign member of many Academies of Sci-
ence (including KNAW, 1978)
Many Prizes (Fields Medal, Prix Gaston Julia, Steele Prize, Wolf
Prize, · · · · · ·)
4
1954, ICM Amsterdam
Hermann Weyl presented the Fields Medals to Kunihiko
Kodaira (1915-1997) and to Serre
5
commentary by Serre (email of 27 December 2004):
“· · · · · · I barely recognize myself on the picture where papy Her-
mann Weyl seems to tell me (and Kodaira): ”Naughty young-
sters! It is OK this time, but don’t do it again ! ” And he gave
my medal to Kodaira, and Kodaira’s medal to me, so that we
had to exchange them the next day.”
6
Fields medal, 3 years after his PhD thesis, for two reasons:
1) The thesis work (introducing ‘spectral sequences’ in algebraic
topology; in particular the ‘Serre spectral sequence’);
2) Introducing ‘sheaf theory’ in complex analytic geometry.
7
(commercial break)
Serre is an interested reader of the 5th Series of Nieuw Archief:
the quote above is part of his reaction
to the 1954 ICM pictures published in
NAW in December 2004;
he sent us several original letters from
him to Alexander Grothendieck, and
from Grothendieck to him, to be used
with a text John Tate wrote for NAW on
the Grothendieck-Serre correspondence
(March 2004)
8
Serre and sports:
apart from skiing and rock climbing, used to be a quite good
table tennis player (but needed an excuse, age difference, when
finally losing from, e.g., Toshiyuki Katsura, 1989, Texel).
9
three conjectures
Serre: “Une conjecture est d’autant plus utile qu’elle est plus
precise, et de ce fait testable sur des exemples.”
11
Serre’s problem on projective modules
1955, problem stated in the paper Faisceaux Algebriques Coherents:
is every projective module M over a polynomial ring R = K[x1, . . . , xn]
(with K a field), free?
(projective means that M is a direct summand of a free module:
M ⊕N ∼= Rn for some module N and integer n)
12
Remark: over many other rings, projective 6= free!
Example 1: R := Z[√−5], M := {a + b
√−5 ∈ R ; a ≡ b mod 2}.
then M is not free;
and R2 ∼= M ⊕M via the map
(f, g) 7→ (2f + (1 +√−5)g, (1−
√−5)f + 2g)
(so M is projective)
14
Example 2, more geometric (Mobius strip):
R := {f ∈ C∞(R) ; f(x + 2π) = f(x)}
(the ring of real C∞-functions on the circle)
M := {m ∈ C∞(R) ; m(x + 2π) = −m(x)}
As in the previous example, M is not free, but R2 ∼= M ⊕M ,
via (f(x), g(x)) 7→(f(x) cos(x/2) + g(x) sin(x/2), f(x) sin(x/2)− g(x) cos(x/2)).
15
Serre’s conjecture on modular forms (1987).
p(x) ∈ Z[x] monic, irreducible,
over C: p(x) =∏(x− αj);
K := Q(α1, . . .) field extension generated by the zeroes of p(x);
Gal(K/Q): the (finite) group of field automorphisms of K;
ρ : Gal(K/Q) ↪→ GL2(Fq) embedding into group of invertible 2×2
matrices over some finite field, with assumptions:
16
1. Take c ∈ Gal(K/Q) complex conjugation restricted to K.
Then det(ρ(c)) = −1 ∈ Fq;
2. ρ is irreducible, i.e., there is no 1-dimensional linear subspace
V ⊂ F2q such that ρ(g) sends V to V for every g ∈ Gal(K/Q).
17
Conjecture (Serre): this situation arises from a modular form.
The work towards understanding this conjecture has been fun-
damental in, e.g., Wiles’ proof of Fermat’s Last Theorem
(work of Ribet, Edixhoven, quite recently Khare, Wintenberger,
Dieulefait)
18
modular form: certain analytic function
H := {z ∈ C ; im(z) > 0} → C,
given by Fourier expansion f(z) = q + a2q2 + . . ., with q = e2πiz,
z ∈ H,
f(az+bcz+d) = ε(d)(cz+d)kf(z) for all
(a bc d
)∈ SL2(Z) with c a multiple
of N
The integer N > 0 is called the level of f ,
the integer k > 0 is called the weight of f
ε : (Z/NZ)∗ → C∗ is called the character of f
19
ρ : Gal(K/Q) ↪→ GL2(Fq) ‘arises from the modular form f ’
means (somewhat imprecise):
there exists ϕ : Z[a2, a3, a4, . . .]→ Fq such that
trace(ρ(Fr`)) = ϕ(a`)
for all but finitely many prime numbers `.
To define Fr`: take splitting field F`n of p(x) mod `;
construct Z[α1, α2, . . .]→ F`n;
‘lift’ the field automorphism ξ 7→ ξ` of F`n to an automorphism
Fr` of Z[α1, α2, . . .] and of the field K.
20
Serre gives a recipe that, given K and ρ, defines a ‘minimal’ level
N (with gcd(N, q) = 1), a minimal weight k, and the character
ε.
In the 90’s Ribet, Mazur, Carayol, Diamond, Edixhoven and oth-
ers proved, that if K and ρ arise from some modular form (with
level coprime to q), then also from one with the level and weight
predicted by Serre.
21
A very simple example:
K is the extension (degree 6) of Q generated by the roots of
x3 − 4x + 4 = 0.
Then K = Q(α,√−11) with α any of the three roots;
Gal(K/Q) ∼= S3 (all permutations of the three roots);
Take any isomorphism ρ : Gal(K/Q)→ GL2(F2)
The pair (K, ρ) arises from the modular form
f(z) = q∞∏
n=1
(1− qn)2(1− q11n)2.
22
this means: write f(z) =∑∞
m=1 amqm =
q−2q2−q3+2q4+q5+2q6−2q7−2q9−2q10+q11−2q12+4q13+. . .
For ` 6= 2, 6= 11 a prime number:
a` is odd⇔
x3 − 4x + 4 is irreducible mod `⇒
` ≡ 1,3,4,5,9 mod 11
23
Also for the primes ` 6= 2, 6= 11:
x3 − 4x + 4 mod ` splits in three linear factors⇔
a` is even & ` ≡ 1,3,4,5,9 mod 11
For odd primes ` ≡ 2,6,7,8,10 mod 11, the number a` is even
and x3 − 4x + 4 mod ` has a linear and a quadratic irreducible
factor.
24
Serre’s conjecture on rational points on curves of genus 3 over
a finite field
(1985, course given at Harvard)
Finite field Fq (for simplicity: q odd).
Curve C of genus 3 over Fq:
either hyperelliptic, which means a complete curve (so, including
two points ‘at infinity’), given by an equation y2 = f(x), with
f(x) a polynomial of degree 8 over Fq without multiple factors;
or a nonsingular curve in P2 given by a quartic equation over Fq.
25
more generally, a nonsingular curve in P2 given by an equation
of degree d has genus g = (d− 1)(d− 2)/2.
The set of points on C with coordinates in Fq is denoted C(Fq).
H. Hasse and A. Weil: #C(Fq) ≤ q + 1 + 2g√
q (here g is the
genus of C)
Improvement by Serre (1985): #C(Fq) ≤ q + 1 + g[2√
q] (here
[x] denotes the largest integer ≤ x)
26
Serre’s conjecture: these bounds are sharp for g = 3, in the
following sense: there should exist an integer ε such that, for
every finite field Fq, a curve C of genus g = 3 over Fq exists with
#C(Fq) ≥ q + 1 + g[2√
q]− ε
The analogous statement for g = 1 is true (M. Deuring, 1940’s)
Serre proves the analogous statement for g = 2 in his Harvard
course (1985)
27
A similar conjecture for g >> 0 cannot be expected: fix q and
compare
limg→∞
q + 1 + g[2√
q]− ε
g= [2
√q]
with a theorem of Drinfeld & Vladut (1983):
lim supg→∞
maxC of genus g #C(Fq)
g≤ √q − 1
28
some results towards this conjecture:
1) Ibukiyama, 1993: restrict to Fq with q odd, q a square butnot a fourth power. Then the conjecture holds, with ε = 0.
2) with R. Auer, 2002: restrict to Fq with q = 3n, all n ≥ 1.Then the conjecture holds, with ε = 21.
3) Serre & Lauter, 2002 and independently Auer & Top, 2002:take ε = 3 (we: ε = 21). For every finite field Fq, there exists acurve C over Fq of genus 3 for which either
#C(Fq) ≥ q + 1 + g[2√
q]− ε
or
#C(Fq) ≤ q + 1− g[2√
q] + ε
(but cannot decide which of the two. . .)
29